<p>In this paper we investigate how the singular flow lines of a Gutierrez–Sotomayor flow are allowed to travel along the ambient two-manifold, monitoring their trajectories as they pass through isolating blocks which compose the manifold. The idea is to register the topological obstructions that may occur depending on what singularities make up the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-limit and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\omega \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation>-limit sets of the singular flow lines. Here we put this approach to test, considering only singular flow lines of cross-cap singularities. We obtain another proof of the realizability of Lyapunov graphs containing only singularities <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\({\mathcal {R}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">R</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\({\mathcal {C}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\({\mathcal {W}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">W</mi> </math></EquationSource> </InlineEquation>. Plus, we characterize all topologically non-equivalent flows associated to such graphs. Surprisingly, the level sets of the respective ambient manifolds can be the same.</p>